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Why simple modules are often finite-dimensional II July 22, 2009

Posted by Akhil Mathew in algebra, representation theory.
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I had a post a few days back on why simple representations of algebras over a field {k} which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.

So, let’s review the notation: {k} is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), {A} is a {k}-algebra, {Z} its center. We assume {Z} is a finitely generated ring over {k}, so in particular Noetherian: each ideal of {Z} is finitely generated.

Theorem 1 (Dixmier, Quillen) If {A} is a finite {Z}-module, then any simple {A}-module is a finite-dimensional {k}-vector space.

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Why simple modules are often finite-dimensional, I July 19, 2009

Posted by Akhil Mathew in algebra.
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Today I want to talk (partially) about a general fact, that first came up as a side remark in the context of my project, and which Dustin Clausen, David Speyer, and I worked out a few days ago.  It was a useful bit of algebra for me to think about.

Theorem 1 Let {A} be an associative algebra with identity over an algebraically closed field {k}; suppose the center {Z \subset A} is a finitely generated ring over {k}, and {A} is a finitely generated {Z}-module. Then: all simple {A}-modules are finite-dimensional {k}-vector spaces.

We’ll get to this after discussing a few other facts about rings, interesting in their own right.

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